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Classwork 3.7b NAME:
Find the general formula for the derivative of y = f(x) =√
(3 + ex).
Then find f ′(0) and and equation for the tangent line at x = 0.
Classwork 3.7a NAME:
Find the general formula for the derivative of y = f(x) = sin32x.
Then find the values of x in [−π, π] such that f ′(x) = 0.
Classwork 3.6 NAME:
1. Find the derivative of f(x) = x cos x.
2. Find the first and second derivatives of g(x) = sin xx .
Classwork 3.5 NAME:
Find the first and second derivatives of the following functions.
1. f(x) = (x2 + x)ex.
2. g(x) = 1x2+1
Classwork 3.4 NAME:
Find how long it takes to stop and how far a car travels when braking from 60
mph to 0 with a velocity v = 60 − 10t and distance traveled s = 60t − 5t2 after t
seconds.
Classwork 3.3 NAME:
Find the following derivatives. Show how the product and/or quotient rules are
used.
1. f(x) = x2 ex.
2. g(x) =√x
x+1 .
Classwork 3.2b NAME:
Compute the derivatives of the following functions, using the power and exponent
rules:
1. f(x) = 3x4 − 2x5.
2. f(x) =√x3
3. f(x) = 2x3 .
Classwork 3.2a NAME:
Use limits to find a general formula for the derivative of f(x) =√x+ 1 at x = a.
Classwork 3.1 NAME:
Let f(x) = x2, so that f(3) = 9.
1. Compute the limit limx→3f(x)−f(3)
x−3 .
2. Observe that the answer to part (1) is the slope of the tangent line to the
curve y = x2 at the point (3, 9). Now find the equation of this tangent line.
Classwork 2.7 NAME:
1. Let f(x) = 2x√x2+4x
. Find the limits limx→−∞f(x) and limx→∞f(x) and the
corresponding horizontal asymptotes.
2. Let f(x) = 36−4x2
x2−16x . Find the limits limx→−∞f(x) and limx→∞f(x) and the
corresponding horizontal asymptotes. Also find the vertical asymptotes.
Classwork 2.6 NAME:
1. Use the Squeeze Theorem to find limx→0x cos(e1x )
2. Evaluate limx→1sin(x−1)x2−1
Classwork 2.4 NAME:
Find limx→0−f(x), limx→0+f(x), limx→0f(x), limx→1−f(x), limx→1+f(x), limx→1f(x),
where
f(x) =
sin x, if x < 0;
x2, if 0 ≤ x ≤ 1;
x+ 1, if x > 1.
Then determine whether f is continuous, left continuous or right continuous, at
x = 0 and/or at x = 1.
Classwork 2.3 NAME:
Evaluate the following limits, if possible:
1. limx→31
x−3 −1
x2−5x+6 .
2. limx→2+ln(x−2)x−3 .
Classwork 2.2 NAME:
1. Use the limit defintion to verify that limx→3 2x− 1 = 5. How close to 3 does
x need to be so that 2x− 1 is within .1 of 5?
2. Find the (possibly infinite) limits limx→2−1
x2−4 and limx→2+1
x2−4 .
Classwork 2.1 NAME:
1. The following table illustrates the growth in the population infected with
the flu virus over time. Find the average rate of change in the infected population
during the intervals [12, 18], [15, 18], and [17, 18]. Estimate the exact rate of change
for day 18.
Days 0 10 12 15 17 18 —
Infected 0 15 23 36 45 50 —
2. The graph below shows the number of Twitter users in Gainesville in recent
years. Find the average increase in the number of users between 2010 and 2014.
Estimate the exact rate of change at time t = 2013 based on the given tangent line.
Classwork 1.6 NAME:
1. Simplify 92/3 · 35/3.
2. Solve 2 ln x+ ln(1 + 1x2 ) = ln 5.
Classwork 1.5 NAME:
1. Find sin−1(sin( 17π3 )) without any calculations.
2. Let f(x) = 3x− 5. Find a formula for the inverse function f−1.
Classwork 1.4 NAME:
1. Given that 0 ≤ θ ≤ π/2 and that tan θ = 2/3, find cos θ and sin θ.
2. Find a value of θ such that cos θ = sin 2θ.
Classwork 1.3 NAME:
1. Find the domain and range of f(t) =√t− 2.
2. Find the composition functions f ◦ g and g ◦ f where
f(t) = 2t+ 3 and g(x) = 1/x.
Classwork 1.2 NAME:
1. Find the slope of the line 2x+ 3y = 5.
Hint: solve for y.
2. Find an equation of the line passing through points (2, 3) and (4, 8).
3. Complete the square and find the minimum value of y = f(x) = 3x2+12x−5.
Classwork 1.1 NAME:
1. Express the set of numbers {x : |2x− 6| < 5} as an interval.
Hint: first simplify the inequality |2x− 6| < 5.
2. Which function is odd/even/neither:
a. f(x) = x2 − 3
b. g(x) = x4 + x3
c. h(x) = 1x3+2x
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